Call Now to Set Up Tutoring:

(888) 888-0446

Complex Analysis : Residue Theory

Study concepts, example questions & explanations for Complex Analysis

Create An Account

All Complex Analysis Resources

1 Diagnostic Test 13 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #11 : Residue Theory

Find the residue at z = 0 for the function

$f(z) = \frac{4}{z+z^2}$ .

Possible Answers:

$\frac{1}{4}$

$\pi$

$\frac{1}{2}$

Correct answer:

Explanation:

Observe,

$f(z) \newline = \frac{4}{z + z^2} \newline =\frac{4}{z} * \frac{1}{z+1} \newline = \frac{4}{z} \displaystyle\sum_{k=0}^{\infty} (-1)^{k} z^{k} \newline = 4\displaystyle\sum_{k=0}^{\infty} (-1)^{k} z^{k -1} \newline = 4(\frac{1}{z} -1 + z - z^2 +z^3 + \cdots)$

The coefficient of $\frac{1}{z}$ is 4*1=4 .

Thus,

$\text{Res}_{z=0} [f(z)] = \text{Res}_{z=0} [\frac{4}{z+z^2}] = 4$ .

Report an Error

Example Question #11 : Residue Theory

Find the residue at z=0 of the function

$f(z) = \frac{z^2-sin(z)}{z^2}$ .

Possible Answers:

$-\frac{1}{2}$

$\frac{7}{9}$

Correct answer:

Explanation:

Observe,

$f(z) \newline = \frac{z-sin(z)}{z} \newline = 1 - \frac{sin(z)}{z^2} \newline = 1 - \displaystyle\sum_{k=0}^{\infty} \frac{-1^{k}}{(2n+1)!} z^{2n-1}$ .

The coefficient of $\frac{1}{z}$ is $\frac{(-1)^0}{(2*0+1)!}= 1$ .

Thus,

$\text{Res}_{z=0} [f(z)] = \text{Res}_{z=0} [\frac{z^2 - sin(z)}{z^2}] = -1$ .

Report an Error

View Tutors

Samantha
Certified Tutor

Middlesex County College, Associate in Science, Sustainability Studies.

View Tutors

Lorie
Certified Tutor

Appalachian State University, Doctorate (PhD), Educational Leadership - Instructional Tech.

View Tutors

Meldorae
Certified Tutor

Wayne State University, Bachelor of Science, Applied Psychology. Wayne State University, Master of Arts Teaching, History.

All Complex Analysis Resources

1 Diagnostic Test 13 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Computer Science Tutors in Atlanta, LSAT Tutors in Philadelphia, Algebra Tutors in San Francisco-Bay Area, GMAT Tutors in Atlanta, Biology Tutors in Seattle, SSAT Tutors in Chicago, Spanish Tutors in Boston, SSAT Tutors in New York City, Calculus Tutors in Houston, French Tutors in Los Angeles

Popular Courses & Classes

Spanish Courses & Classes in San Diego, GMAT Courses & Classes in Washington DC, ACT Courses & Classes in Washington DC, Spanish Courses & Classes in Boston, Spanish Courses & Classes in Atlanta, ISEE Courses & Classes in Phoenix, GRE Courses & Classes in Denver, ACT Courses & Classes in Seattle, ISEE Courses & Classes in New York City, MCAT Courses & Classes in San Diego

Popular Test Prep

GMAT Test Prep in Washington DC, GRE Test Prep in Miami, GRE Test Prep in Denver, MCAT Test Prep in Los Angeles, LSAT Test Prep in New York City, SSAT Test Prep in Miami, ACT Test Prep in Seattle, GRE Test Prep in Atlanta, LSAT Test Prep in Dallas Fort Worth, GRE Test Prep in Houston

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Complex Analysis : Residue Theory

Study concepts, example questions & explanations for Complex Analysis

All Complex Analysis Resources

Example Questions

Example Question #11 : Residue Theory

Example Question #11 : Residue Theory

All Complex Analysis Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: